View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Algebra 319 (2008) 759–776 www.elsevier.com/locate/jalgebra Bounds for finite primitive complex linear groups Michael J. Collins a,b a University College, Oxford OX1 4BH, England, UK b Department of Mathematics, Ohio State University, Columbus, OH 43210, USA 1 Received 17 August 2005 Available online 1 February 2007 Communicated by Ronald Solomon In memoriam Walter Feit 1930–2004 Abstract In 1878, Jordan showed that a finite complex linear group must possess a normal abelian subgroup whose index is bounded by a function of the degree n alone. In this paper, we study primitive groups; when n>12, the optimal bound is (n + 1)!, achieved by the symmetric group of degree n + 1. We obtain the optimal bounds in smaller degree also. Our proof uses known lower bounds for the degrees of the faithful representations of each quasisimple group, for which the classification of finite simple groups is required. In a subsequent paper [M.J. Collins, On Jordan’s theorem for complex linear groups, J. Group Theory 10 (2007) 411–423] we will show that (n + 1)! is the optimal bound in general for Jordan’s theorem when n 71. © 2006 Elsevier Inc. All rights reserved. Keywords: Finite primitive complex linear groups 1. Introduction It is well known that the complex representation of the symmetric group Sn of degree n − 1 that occurs as a constituent of the standard permutation representation has the smallest degree amongst all its faithful representations in characteristic 0; equivalently, Sn+1 is the largest sym- metric group that can be embedded in the matrix group GL(n, C). In a paper directed towards E-mail address: [email protected]. 1 This paper was written while the author held a Visiting Professorship, partially supported by the Mathematics Research Institute of the Ohio State University. 0021-8693/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jalgebra.2005.11.042 760 M.J. Collins / Journal of Algebra 319 (2008) 759–776 an application to linear differential equations [13], Jordan showed that, modulo an abelian nor- mal subgroup, the order of a finite group embedded in GL(n, C) can be bounded by a function of n. Jordan did not give any explicit bound, establishing only existence by induction on n; later Frobenius [11] and Schur [16] gave explicit bounds. These, though, prove to be of orders of magnitude that we will show in the final section are far too large. Here we will show, using the classification of finite simple groups, that the group Sn+1 does in fact determine the correct bound generically for primitive groups; in a subsequent paper [6] we will extend this result to drop the assumption of primitivity while in a third paper [7] we will look at analogues for representations over fields of nonzero characteristic. Previously, Boris Weisfeiler [21] had announced a result that gave bounds of approximately the right asymptotic order of magnitude where he too assumed the classification. Sadly he dis- appeared in 1985, leaving a near-complete manuscript in which he obtained a result close to that which we will obtain — since he was interested also in the corresponding problem for linear groups in nonzero characteristic p, he needed to allow for arbitrarily large groups of Lie type in that characteristic, and also for the fact that Sn has an irreducible representation of degree n − 2 when p divides n and so gave a generic bound of (n + 2)!. Some years ago, Walter Feit asked me to prepare Weisfeiler’s work for publication. However, while Weisfeiler relied heav- ily on studying some quite delicate functions in order to obtain overarching bounds (and some of that work is in a missing appendix), a deeper analysis of the group theoretic structure of, in particular, primitive groups allows for a proper understanding of what is happening, and for a very precise description of the small obstructions to the “generic bound”; that part dealing with primitive groups will be presented here. Much of the technical argument lies in determining lower bounds for the degrees of faithful projective representations for every finite simple group, and this is where the classification is used; this work has been carried out by others although we should remark that, except for a detailed analysis of small cases, quite crude bounds would actually suffice for our purpose, and only for the convenience of their expression will we use more recently obtained results in place of those already available to Weisfeiler. Prior to the classification of finite simple groups, Feit himself had studied primitive complex linear groups of small degree. Jordan in his original paper had given a list of the finite linear groups of degrees 2 and 3, and Blichfeldt gave the primitive groups of degree 4 [4].2 Using methods based on modular representation theory and studying specifically groups of prime de- gree, Brauer had determined the finite linear groups of degree 5, followed by Wales’ thesis on groups of degree 7 and Lindsay’s on groups of degree 6; Feit then worked on groups of degrees 8, 9 and 10. (See [10] and references cited therein.) Interestingly, the highest degree studied in that “programme” was for n = 11 by Robinson in his thesis [15] when, as we will see, the final obstruction is the pair of dual 12-dimensional representations of the 6-fold cover of Suzuki’s sporadic simple group. Our goal in this paper is to establish the generic bound for primitive complex groups, and to describe the small exceptions. In particular, it will be seen that the generic bound for primitive complex groups will be satisfied as soon as n>12, and Feit used this in unpublished work on representations over cyclotomic fields. In the process, we will obtain some general structure theorems that are independent of the characteristic of the representation so that these may be applied when we later consider imprim- 2 Though both Jordan and Blichfeldt missed groups. M.J. Collins / Journal of Algebra 319 (2008) 759–776 761 itive groups and groups of nonzero characteristic [6,7]. Indeed, we will see then that the generic bound that we obtain here holds for all complex groups provided that n 71. Recall that (in arbitrary characteristic) an irreducible representation of a group G is said to be primitive if the underlying vector space cannot be decomposed as a direct sum of proper subspaces permuted under the action of G. It follows from Nakayama’s generalisation of the Frobenius reciprocity theorem (see, for example, [5, Chapter 1, Theorem 35]) that an irreducible representation is primitive if and only if it is not induced from that of any proper subgroup. We will say that the group G is primitive when G is a subgroup of GL(n, k) and the representation so afforded is primitive. Theorem A. Let G be a finite primitive subgroup of GL(n, C) and suppose that n>1. Then ∼ ∼ [G : Z(G)] is bounded. If the bound is achieved, then G = An+1 and G/Z(G) = Sn+1, with the following exceptions: n [G : Z(G)][G : Z(G)]/(n + 1)! H 26010 2.A5 3 360 15 3.A6 4 25920 216 Sp4(3) 5 25920 36 PSp4(3) 6 6531840 1296 61.U4(3).22 7 1451520 36 Sp6(2) + 8 348368800 960 2.O8 (2).2 1+4 9 4199040 1.157 3 .Sp4(3) 12 448345497600 72 6.Suz where H is a group uniquely determined up to isoclinism,3 and G = Z(G).H. In particular, in general [G : Z(G)]!(n + 1)! if n>12. Here, in the final column, we have followed the ATLAS [9] to describe the normal structure of a group H with |Z(H)| minimal and to describe which extensions (central, or by an auto- morphism) occur. We have used the notation of the tables in the ATLAS for the simple groups, except for Sp and PSp as the symplectic groups and their simple quotients since they arise with their natural actions in Sections 2 and 3. We will also use “full” linear notation when discussing + classical groups in Section 4. However, we note that here O8 (2) is the simple group more often + = + written as Ω8 (2)( O (8, 2) ). (See also Chapter 2 of [9].) One may reasonably ask why one gets these small exceptions, other than by small numeri- cal accident. While this undoubtedly has a role, it would appear that the answer may be partly geometric too. The generic examples of the symmetric groups are, of course, Coxeter groups and hence groups generated by reflections, but we observe too that the Weyl group W(E8) has a + =∼ × =∼ normal structure 2.O8 (2).2, while W(E7) Z2 Sp6(2) and W(E6) Aut(PSp4(3)). Further- more, the stated groups in degrees 4, 5 and 6 may be realised as complex reflection groups, while Z2 × 3.A6 also occurs as a (3-dimensional) complex reflection group. ∼ The isomorphism W(E6) = Aut(PSp4(3)) has a further interesting consequence. The group U4(3) contains two conjugacy classes of maximal subgroups isomorphic to PSp4(3), and these → fuse under an outer automorphism of order 4, with an embedding Aut(PSp4(3)) U4(3).22. 3 This is relevant only in degrees 6 and 8, where there are two possible extensions of the derived group, and either may be taken. Recall that two groups that are isoclinic possess central extensions that are isomorphic.